Neville theta functions

{{inuse} Neville Theta Functions are defined as follow:

$NevilleC(z,m)={\frac {{\sqrt {(}}2)*q(m)^{1/4}*(\sum _{k=0}^{\infty }(q(m)^{(}k*(k+1))*cos((1/2)*(2*k+1)*\pi *z/K(m))))}{{\sqrt {(}}K(m))*m^{1/4}}}$ $NevilleThetaC(z,m)={\frac {{\sqrt {(}}2*\pi )*q(m)^{1/4}*(\sum _{k=0}^{\infty }(q(m)^{k*(k+1)}*cos((1/2)*(2*k+1)*\pi *z/K(m))))}{{\sqrt {(}}K(m))*m^{1/4}}}$ $NevilleThetaD(z,m)={\frac {{\sqrt {(}}(1/2)*\pi )*(1+2*(\sum _{k=1}^{\infty }(q(m)^{(}k^{2})*cos(k*\pi *z/K(m)))))}{{\sqrt {(}}K(m))}}$ $NevilleThetaN(z,m)={\frac {{\sqrt {(}}\pi )*(1+2*(\sum _{k=1}^{\infty }((-1)^{k}*q(m)^{k^{2}}*cos(k*\pi *z/K(m)))))}{{\sqrt {(}}2)*(1-m)^{(}1/4)*{\sqrt {K(m)}}}}$

where:

$K(m)=EllipticK({\sqrt {(}}m))$
$K'(m)=EllipticK({\sqrt {(}}1-m))$
$q(m)=e^{\frac {-\pi *K(m)}{K'(m)}}$ is the elliptic nome

Examples

$NevilleThetaC(2.5,.3)=-.65900466676738154967$
$NevilleThetaD(2.5,.3)=0.95182196661267561994$
$NevilleThetaN(2.5,.3)=1.0526693354651613637$
$NevilleThetaS(2.5,.3)=0.82086879524530400536$

Symmetry

$NevilleThetaC(z,m)=NevilleThetaC(-z,m)$
$NevilleThetaD(z,m)=NevilleThetaD(-z,m)$
$NevilleThetaN(z,m)=NevilleThetaN(-z,m)$
$NevilleThetaS(z,m)=-NevilleThetaS(-z,m)$

Series expansion

$NevilleThetaC(z,1/2)=.9998-.3641*z^{2}+0.2466e-1*z^{4}-0.1210e-2*z^{6}+0.8707e-4*z^{8}+O(z^{1}0)$
$NevilleThetaD(z,1/2)=.9995-.1143*z^{2}+0.2736e-1*z^{4}-0.2629e-2*z^{6}+0.1368e-3*z^{8}+O(z^{1}0)$
$NevilleThetaN(z,1/2)=1.000+.1358*z^{2}-0.3244e-1*z^{4}+0.3093e-2*z^{6}-0.1561e-3*z^{8}+O(z^{1}0)$
$NevilleThetaS(z,1/2)=1.000*z-.1142*z^{3}+0.2358e-2*z^{5}+0.2276e-3*z^{7}-0.2630e-4*z^{9}+O(z^{1}1)$